Localization of eigenvalues of doubly cyclic matrices

C. Johnson, Z. Price, and I. Spitkovsky conjectured that in this family, the number of eigenvalues in the left half-plane is maximized by αI−βΣ⁎; we prove this conjecture. Moreover, the complete range of possibilities for the number of eigenvalues in the left half-plane is demonstrated: if α<β, then any odd number between 1 and the maximum, inclusive, is attainable.